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authorKarl Berry <karl@freefriends.org>2020-02-03 22:30:53 +0000
committerKarl Berry <karl@freefriends.org>2020-02-03 22:30:53 +0000
commitc968dc6a6e3b376bbba491ffad313ee8e82dab04 (patch)
tree11eb84dac11ad9b601a5c77745af1ebcb85dd2af /Master/texmf-dist/doc
parent80eeb8a04335539fde455e00d5cfde4e2c7ed5c6 (diff)
lie-hasse (3feb20)
git-svn-id: svn://tug.org/texlive/trunk@53653 c570f23f-e606-0410-a88d-b1316a301751
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-rw-r--r--Master/texmf-dist/doc/latex/lie-hasse/lie-hasse.bib495
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@@ -0,0 +1,21 @@
+___________________________________
+
+ Lie Hasse
+
+ v1.0
+
+ 3 February 2020
+___________________________________
+
+Authors : Ben McKay
+Maintainer: Ben McKay
+E-mail : b.mckay@ucc.ie
+Licence : Released under the LaTeX Project Public License v1.3c or
+ later, see http://www.latex-project.org/lppl.txt
+
+----------------------------------------------------------------------
+
+This package draws Hasse diagrams of the posets
+of the simple roots of any complex simple Lie algebra.
+It uses the Dynkin diagrams package dynkin-diagrams.
+
diff --git a/Master/texmf-dist/doc/latex/lie-hasse/lie-hasse.bib b/Master/texmf-dist/doc/latex/lie-hasse/lie-hasse.bib
new file mode 100644
index 00000000000..d39af91eae2
--- /dev/null
+++ b/Master/texmf-dist/doc/latex/lie-hasse/lie-hasse.bib
@@ -0,0 +1,495 @@
+% Encoding: ISO-8859-1
+
+
+@Book{Adams:1996,
+ Title = {Lectures on exceptional {L}ie groups},
+ Author = {Adams, J. F.},
+ Publisher = {University of Chicago Press, Chicago, IL},
+ Year = {1996},
+ Note = {With a foreword by J. Peter May,
+ Edited by Zafer Mahmud and Mamoru Mimura},
+ Series = {Chicago Lectures in Mathematics},
+
+ ISBN = {0-226-00526-7; 0-226-00527-5},
+ Mrclass = {22-01 (22E10)},
+ Mrnumber = {1428422},
+ Mrreviewer = {William M. McGovern},
+ Owner = {user},
+ Pages = {xiv+122},
+ Timestamp = {2018.07.22}
+}
+
+@Article{Baba:2009,
+ Title = {Satake diagrams and restricted root systems of semisimple pseudo-{R}iemannian symmetric spaces},
+ Author = {Baba, Kurando},
+ Journal = {Tokyo J. Math.},
+ Year = {2009},
+ Number = {1},
+ Pages = {127--158},
+ Volume = {32},
+
+ Fjournal = {Tokyo Journal of Mathematics},
+ ISSN = {0387-3870},
+ Mrclass = {17B20 (17B22 53C35)},
+ Mrnumber = {2541161},
+ Mrreviewer = {Oksana S. Yakimova},
+ Owner = {user},
+ Timestamp = {2017.12.04},
+ Url = {https://doi.org/10.3836/tjm/1249648414}
+}
+
+@Book{Bourbaki:2002,
+ Title = {Lie groups and {L}ie algebras. {C}hapters 4--6},
+ Author = {Bourbaki, Nicolas},
+ Publisher = {Springer-Verlag, Berlin},
+ Year = {2002},
+ Note = {Translated from the 1968 French original by Andrew Pressley},
+ Series = {Elements of Mathematics (Berlin)},
+
+ ISBN = {3-540-42650-7},
+ Mrclass = {17-01 (00A05 20E42 20F55 22-01)},
+ Mrnumber = {1890629},
+ Owner = {user},
+ Pages = {xii+300},
+ Timestamp = {2017.11.15},
+ Url = {https://doi.org/10.1007/978-3-540-89394-3}
+}
+
+@Book{Carter:2005,
+ Title = {Lie algebras of finite and affine type},
+ Author = {Carter, R. W.},
+ Publisher = {Cambridge University Press, Cambridge},
+ Year = {2005},
+ Series = {Cambridge Studies in Advanced Mathematics},
+ Volume = {96},
+
+ ISBN = {978-0-521-85138-1; 0-521-85138-6},
+ Mrclass = {17-02 (17B67)},
+ Mrnumber = {2188930},
+ Mrreviewer = {Stephen Slebarski},
+ Owner = {user},
+ Pages = {xviii+632},
+ Timestamp = {2017.11.15},
+ Url = {https://doi.org/10.1017/CBO9780511614910}
+}
+
+@InCollection{Carter:1995,
+ Title = {On the representation theory of the finite groups of {L}ie
+ type over an algebraically closed field of characteristic 0 [
+ {MR}1170353 (93j:20034)]},
+ Author = {Carter, R. W.},
+ Booktitle = {Algebra, {IX}},
+ Publisher = {Springer, Berlin},
+ Year = {1995},
+ Pages = {1--120, 235--239},
+ Series = {Encyclopaedia Math. Sci.},
+ Volume = {77},
+
+ Doi = {10.1007/978-3-662-03235-0_1},
+ Mrclass = {20C33 (20-02 20G05)},
+ Mrnumber = {1392478},
+ Owner = {user},
+ Timestamp = {2018.05.19},
+ Url = {https://doi.org/10.1007/978-3-662-03235-0_1}
+}
+
+@Article{Chuah:2013,
+ Title = {Cartan automorphisms and {V}ogan superdiagrams},
+ Author = {Chuah, Meng-Kiat},
+ Journal = {Math. Z.},
+ Year = {2013},
+ Number = {3-4},
+ Pages = {793--800},
+ Volume = {273},
+
+ Fjournal = {Mathematische Zeitschrift},
+ ISSN = {0025-5874},
+ Mrclass = {17B20 (17B40)},
+ Mrnumber = {3030677},
+ Mrreviewer = {Zi-Xin Hou},
+ Owner = {user},
+ Timestamp = {2017.12.04},
+ Url = {https://doi.org/10.1007/s00209-012-1030-z}
+}
+
+@InCollection{Draper/Guido:2016,
+ Title = {On the real forms of the exceptional {L}ie algebra {$\mathfrak
+ e_6$} and their {S}atake diagrams},
+ Author = {Draper Fontanals, Cristina and Guido, Valerio},
+ Booktitle = {Non-associative and non-commutative algebra and operator
+ theory},
+ Publisher = {Springer, Cham},
+ Year = {2016},
+ Pages = {211--226},
+ Series = {Springer Proc. Math. Stat.},
+ Volume = {160},
+
+ Mrclass = {17B20 (17A75 17B25 17B60)},
+ Mrnumber = {3613831},
+ Mrreviewer = {Alberto Elduque},
+ Owner = {user},
+ Timestamp = {2018.04.30}
+}
+
+@Book{Dynkin:2000,
+ Title = {Selected papers of {E}. {B}. {D}ynkin with commentary},
+ Author = {Dynkin, E. B.},
+ Publisher = {American Mathematical Society, Providence, RI; International Press, Cambridge, MA},
+ Year = {2000},
+ Note = {Edited by A. A. Yushkevich, G. M. Seitz and A. L. Onishchik},
+
+ ISBN = {0-8218-1065-0},
+ Mrclass = {01A75 (60Jxx)},
+ Mrnumber = {1757976},
+ Mrreviewer = {William M. McGovern},
+ Owner = {user},
+ Pages = {xxviii+796},
+ Timestamp = {2017.11.15}
+}
+
+@Article{Dynkin:1952,
+ Title = {Semisimple subalgebras of semisimple {L}ie algebras},
+ Author = {Dynkin, E. B.},
+ Journal = {Mat. Sbornik N.S.},
+ Year = {1952},
+ Note = {Reprinted in English translation in \cite{Dynkin:2000}.},
+ Pages = {349--462 (3 plates)},
+ Volume = {30(72)},
+
+ Mrclass = {09.1X},
+ Mrnumber = {0047629},
+ Mrreviewer = {I. Kaplansky},
+ Owner = {user},
+ Timestamp = {2017.11.15}
+}
+
+@Article{Frappat/Sciarrino/Sorba:1989,
+ Title = {Structure of basic {L}ie superalgebras and of their affine extensions},
+ Author = {Frappat, L. and Sciarrino, A. and Sorba, P.},
+ Journal = {Comm. Math. Phys.},
+ Year = {1989},
+ Number = {3},
+ Pages = {457--500},
+ Volume = {121},
+
+ Fjournal = {Communications in Mathematical Physics},
+ ISSN = {0010-3616},
+ Mrclass = {17B70 (17A70 17B40)},
+ Mrnumber = {990776},
+ Mrreviewer = {A. Pianzola},
+ Owner = {user},
+ Timestamp = {2017.12.18},
+ Url = {http://0-projecteuclid.org.library.ucc.ie/euclid.cmp/1104178142}
+}
+
+@Book{Grove/Benson:1985,
+ Title = {Finite reflection groups},
+ Author = {Grove, L. C. and Benson, C. T.},
+ Publisher = {Springer-Verlag, New York},
+ Year = {1985},
+ Edition = {Second},
+ Series = {Graduate Texts in Mathematics},
+ Volume = {99},
+
+ ISBN = {0-387-96082-1},
+ Mrclass = {20-01 (20B25 20H15)},
+ Mrnumber = {777684},
+ Owner = {user},
+ Pages = {x+133},
+ Timestamp = {2017.11.15},
+ Url = {https://doi.org/10.1007/978-1-4757-1869-0}
+}
+
+@Book{Helgason:2001,
+ Title = {Differential geometry, {L}ie groups, and symmetric spaces},
+ Author = {Helgason, Sigurdur},
+ Publisher = {American Mathematical Society, Providence, RI},
+ Year = {2001},
+ Note = {Corrected reprint of the 1978 original},
+ Series = {Graduate Studies in Mathematics},
+ Volume = {34},
+
+ ISBN = {0-8218-2848-7},
+ Mrclass = {53C35 (22E10 22E46 22E60)},
+ Mrnumber = {1834454},
+ Owner = {user},
+ Pages = {xxvi+641},
+ Timestamp = {2017.11.15},
+ Url = {https://doi.org/10.1090/gsm/034}
+}
+
+@Book{Humphreys:1990,
+ Title = {Reflection groups and {C}oxeter groups},
+ Author = {Humphreys, James E.},
+ Publisher = {Cambridge University Press, Cambridge},
+ Year = {1990},
+ Series = {Cambridge Studies in Advanced Mathematics},
+ Volume = {29},
+
+ ISBN = {0-521-37510-X},
+ Mrclass = {20-02 (20F32 20F55 20G15 20H15)},
+ Mrnumber = {1066460},
+ Mrreviewer = {Louis Solomon},
+ Owner = {user},
+ Pages = {xii+204},
+ Timestamp = {2017.11.15},
+ Url = {https://doi.org/10.1017/CBO9780511623646}
+}
+
+@Book{Kac:1990,
+ Title = {Infinite-dimensional {L}ie algebras},
+ Author = {Kac, Victor G.},
+ Publisher = {Cambridge University Press, Cambridge},
+ Year = {1990},
+ Edition = {Third},
+
+ ISBN = {0-521-37215-1; 0-521-46693-8},
+ Mrclass = {17B65 (17B67 17B68 58F07)},
+ Mrnumber = {1104219},
+ Owner = {user},
+ Pages = {xxii+400},
+ Timestamp = {2017.11.15},
+ Url = {https://doi.org/10.1017/CBO9780511626234}
+}
+
+@Article{Khastgir/Sasaki:1996,
+ Title = {Non-canonical folding of {D}ynkin diagrams and reduction of affine {T}oda theories},
+ Author = {Khastgir, S. Pratik and Sasaki, Ryu},
+ Journal = {Progr. Theoret. Phys.},
+ Year = {1996},
+ Number = {3},
+ Pages = {503--518},
+ Volume = {95},
+
+ Fjournal = {Progress of Theoretical Physics},
+ ISSN = {0033-068X},
+ Mrclass = {81T10 (17B81 58F07 81R10)},
+ Mrnumber = {1388245},
+ Mrreviewer = {Mehmet Koca},
+ Owner = {user},
+ Timestamp = {2017.12.18},
+ Url = {https://doi.org/10.1143/PTP.95.503}
+}
+
+@book {Langlands:1967,
+ AUTHOR = {Langlands, Robert P.},
+ TITLE = {Euler products},
+ NOTE = {A James K. Whittemore Lecture in Mathematics given at Yale
+ University, 1967,
+ Yale Mathematical Monographs, 1},
+ PUBLISHER = {Yale University Press, New Haven, Conn.-London},
+ YEAR = {1971},
+ PAGES = {v+53},
+ MRCLASS = {10D20 (22E55)},
+ MRNUMBER = {0419366},
+MRREVIEWER = {Stephen Gelbart},
+}
+@MISC {MathOverflow:123801,
+TITLE = {Invariant subbundles of tangent bundle of flag variety (question)},
+AUTHOR = {Benjamin McKay},
+HOWPUBLISHED = {MathOverflow},
+DATE = {2013-03-06},
+NOTE = {URL:\url{http://mathoverflow.net/a/22350} (visited on 2020-01-29)},
+URL = {http://mathoverflow.net/a/123801},
+URLDATE = {2020-01-29},
+}
+@phdthesis{Nutma:2010,
+ author = {Nutma, Teake Aant},
+ title = {Kac-Moody symmetries and gauged supergravity},
+ school = {Rijksuniversiteit Groningen},
+ year = 2010,
+ address = {Groningen},
+ month = 9,
+ note = {URL:\url{http://inspirehep.net/record/1283406/files/Thesis-2010-Nutma.pdf} (visited on 2020-01-29)}
+}
+@Book{OnishchikVinberg:1990,
+ Title = {Lie groups and algebraic groups},
+ Author = {Onishchik, A. L. and Vinberg, {\`E}. B.},
+ Publisher = {Springer-Verlag},
+ Year = {1990},
+
+ Address = {Berlin},
+ Note = {Translated from the Russian and with a preface by D. A. Leites},
+ Series = {Springer Series in Soviet Mathematics},
+
+ ISBN = {3-540-50614-4},
+ Mrclass = {22-01 (17B20 20G20 22E10 22E15)},
+ Mrnumber = {91g:22001},
+ Mrreviewer = {James E. Humphreys},
+ Owner = {user},
+ Pages = {xx+328},
+ Timestamp = {2017.11.15}
+}
+
+@Book{Onishchik/Vinberg:1990,
+ Title = {Lie groups and algebraic groups},
+ Author = {Onishchik, A. L. and Vinberg, \`E. B.},
+ Publisher = {Springer-Verlag, Berlin},
+ Year = {1990},
+ Note = {Translated from the Russian and with a preface by D. A. Leites},
+ Series = {Springer Series in Soviet Mathematics},
+
+ ISBN = {3-540-50614-4},
+ Mrclass = {22-01 (17B20 20G20 22E10 22E15)},
+ Mrnumber = {1064110},
+ Mrreviewer = {James E. Humphreys},
+ Owner = {user},
+ Pages = {xx+328},
+ Timestamp = {2017.11.15},
+ Url = {https://doi.org/10.1007/978-3-642-74334-4}
+}
+@Article{Ransingh:2013,
+ Title = {Vogan diagrams of untwisted affine {K}ac-{M}oody superalgebras},
+ Author = {Ransingh, Biswajit},
+ Journal = {Asian-Eur. J. Math.},
+ Year = {2013},
+ Number = {4},
+ Pages = {1350062, 10},
+ Volume = {6},
+
+ Fjournal = {Asian-European Journal of Mathematics},
+ ISSN = {1793-5571},
+ Mrclass = {17B67 (17B05 17B22 17B40)},
+ Mrnumber = {3149279},
+ Mrreviewer = {Xiangqian Guo},
+ Owner = {user},
+ Timestamp = {2018.01.11}
+}
+
+@Article{Ransingh:unpub,
+ Title = {{Vogan diagrams of affine twisted Lie superalgebras}},
+ Author = {Ransingh, B.},
+ Journal = {ArXiv e-prints},
+ Year = {2013},
+
+ Month = mar,
+ Pages = {1--9},
+
+ Adsnote = {Provided by the SAO/NASA Astrophysics Data System},
+ Adsurl = {http://adsabs.harvard.edu/abs/2013arXiv1303.0092R},
+ Archiveprefix = {arXiv},
+ Eprint = {1303.0092},
+ Keywords = {Mathematical Physics, Mathematics - Representation Theory},
+ Owner = {user},
+ Primaryclass = {math-ph},
+ Timestamp = {2018.01.11}
+}
+
+@Article{Regelskis/Vlaar:2016,
+ Title = {{Reflection matrices, coideal subalgebras and generalized Satake diagrams of affine type}},
+ Author = {{Regelskis}, V. and {Vlaar}, B.},
+ Journal = {ArXiv e-prints},
+ Year = {2016},
+
+ Month = feb,
+ Pages = {1--118},
+
+ Adsnote = {Provided by the SAO/NASA Astrophysics Data System},
+ Adsurl = {http://adsabs.harvard.edu/abs/2016arXiv160208471R},
+ Archiveprefix = {arXiv},
+ Eprint = {1602.08471},
+ Keywords = {Mathematical Physics, Mathematics - Quantum Algebra, Mathematics - Representation Theory, Nonlinear Sciences - Exactly Solvable and Integrable Systems},
+ Owner = {user},
+ Primaryclass = {math-ph},
+ Timestamp = {2017.12.04}
+}
+@ARTICLE{Ringel:2013,
+ author = {Ringel, Claus Michael},
+ title = {The root posets and their rich antichains},
+ journal = {arXiv e-prints},
+ keywords = {Mathematics - Combinatorics, Mathematics - Representation Theory},
+ year = "2013",
+ month = "Jun",
+ eid = {arXiv:1306.1593},
+ pages = {arXiv:1306.1593},
+archivePrefix = {arXiv},
+ eprint = {1306.1593},
+ primaryClass = {math.CO},
+ adsurl = {https://ui.adsabs.harvard.edu/abs/2013arXiv1306.1593R},
+ adsnote = {Provided by the SAO/NASA Astrophysics Data System}
+}
+@Book{Satake:1980,
+ Title = {Algebraic structures of symmetric domains},
+ Author = {Satake, Ichir\^o},
+ Publisher = {Iwanami Shoten, Tokyo; Princeton University Press, Princeton, N.J.},
+ Year = {1980},
+ Series = {Kan\^o Memorial Lectures},
+ Volume = {4},
+
+ Mrclass = {32-02 (17C35 32Mxx 53C35)},
+ Mrnumber = {591460},
+ Mrreviewer = {S. Murakami},
+ Owner = {user},
+ Pages = {xvi+321},
+ Timestamp = {2017.11.15}
+}
+
+@Book{Springer:2009,
+ Title = {Linear algebraic groups},
+ Author = {Springer, T. A.},
+ Publisher = {Birkh\"auser Boston, Inc., Boston, MA},
+ Year = {2009},
+ Edition = {second},
+ Series = {Modern Birkh\"auser Classics},
+
+ ISBN = {978-0-8176-4839-8},
+ Mrclass = {20G15 (14L10)},
+ Mrnumber = {2458469},
+ Owner = {user},
+ Pages = {xvi+334},
+ Timestamp = {2018.03.31}
+}
+
+@InCollection{Zuber:1998,
+ Title = {Generalized {D}ynkin diagrams and root systems and their folding},
+ Author = {Zuber, Jean-Bernard},
+ Booktitle = {Topological field theory, primitive forms and related topics ({K}yoto, 1996)},
+ Publisher = {Birkh\"auser Boston, Boston, MA},
+ Year = {1998},
+ Pages = {453--493},
+ Series = {Progr. Math.},
+ Volume = {160},
+
+ Mrclass = {17B20 (05C25 20F55)},
+ Mrnumber = {1653035},
+ Mrreviewer = {Saeid Azam},
+ Owner = {user},
+ Timestamp = {2017.12.18}
+}
+
+@Book{Vinberg:1994,
+ Title = {Lie groups and {L}ie algebras, {III}},
+ Editor = {Vinberg, \`E. B.},
+ Publisher = {Springer-Verlag, Berlin},
+ Year = {1994},
+ Note = {Structure of Lie groups and Lie algebras, A translation of {{\i}t Current problems in mathematics. Fundamental directions. Vol. 41} (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1990 [ MR1056485 (91b:22001)], Translation by V. Minachin [V. V. Minakhin], Translation edited by A. L. Onishchik and \`E. B. Vinberg},
+ Series = {Encyclopaedia of Mathematical Sciences},
+ Volume = {41},
+
+ ISBN = {3-540-54683-9},
+ Mrclass = {22-06 (17-06 22Exx)},
+ Mrnumber = {1349140},
+ Owner = {user},
+ Pages = {iv+248},
+ Timestamp = {2017.11.15},
+ Url = {https://doi.org/10.1007/978-3-662-03066-0}
+}
+
+@Book{Fulton.Harris:1991,
+ title = {Representation theory},
+ publisher = {Springer-Verlag, New York},
+ year = {1991},
+ author = {Fulton, William and Harris, Joe},
+ volume = {129},
+ series = {Graduate Texts in Mathematics},
+ isbn = {0-387-97527-6; 0-387-97495-4},
+ note = {A first course, Readings in Mathematics},
+ doi = {10.1007/978-1-4612-0979-9},
+ mrclass = {20G05 (17B10 20G20 22E46)},
+ mrnumber = {1153249},
+ mrreviewer = {James E. Humphreys},
+ pages = {xvi+551},
+ url = {https://doi.org/10.1007/978-1-4612-0979-9},
+}
+
+@Comment{jabref-meta: databaseType:bibtex;}
diff --git a/Master/texmf-dist/doc/latex/lie-hasse/lie-hasse.pdf b/Master/texmf-dist/doc/latex/lie-hasse/lie-hasse.pdf
new file mode 100644
index 00000000000..5be5791e7ff
--- /dev/null
+++ b/Master/texmf-dist/doc/latex/lie-hasse/lie-hasse.pdf
Binary files differ
diff --git a/Master/texmf-dist/doc/latex/lie-hasse/lie-hasse.tex b/Master/texmf-dist/doc/latex/lie-hasse/lie-hasse.tex
new file mode 100644
index 00000000000..82643fee141
--- /dev/null
+++ b/Master/texmf-dist/doc/latex/lie-hasse/lie-hasse.tex
@@ -0,0 +1,555 @@
+\documentclass{amsart}
+\title[The Lie Hasse package]{The Lie Hasse package \\ Version 1.0}
+%% My name:
+\makeatletter
+\DeclareRobustCommand{\scotsMc}{\scotsMcx{c}}
+\DeclareRobustCommand{\scotsMC}{\scotsMcx{\textsc{c}}}
+\DeclareRobustCommand{\scotsMcx}[1]{%
+ M%
+ \raisebox{\dimexpr\fontcharht\font`M-\height}{%
+ \check@mathfonts\fontsize{\sf@size}{0}\selectfont
+ \kern.3ex\underline{\kern-.3ex #1\kern-.3ex}\kern.3ex
+ }%
+}
+\expandafter\def\expandafter\@uclclist\expandafter{%
+ \@uclclist\scotsMc\scotsMC
+}
+\makeatother
+\newcommand{\authorsname}{\texorpdfstring{Benjamin \scotsMc{}Kay}{Benjamin McKay}}
+\author{\authorsname}
+\address{School of Mathematical Sciences, University College Cork, Cork, Ireland}
+\email{b.mckay@ucc.ie}
+\date{3 February 2020}
+\usepackage{etex}
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenx}
+\usepackage{etoolbox}
+\usepackage{lmodern}
+\RequirePackage[tt=lining]{cfr-lm}
+\usepackage[kerning=true,tracking=true]{microtype}
+\usepackage{amsmath}
+\usepackage{amsfonts}
+\usepackage{mathtools}
+\usepackage{mathtext}
+\usepackage[english]{babel}
+\usepackage[pagebackref]{hyperref}
+ \hypersetup{
+ colorlinks = true, %Colours links instead of ugly boxes
+ urlcolor = black, %Colour for external hyperlinks
+ linkcolor = black, %Colour of internal links
+ citecolor = black %Colour of citations
+ }
+\usepackage{lie-hasse}
+\usetikzlibrary{positioning}
+\usepackage{fancyvrb}\fvset{obeytabs,tabsize=2,fontsize=\small}
+\usepackage[listings]{tcolorbox}
+\tcbuselibrary{breakable}
+\tcbuselibrary{skins}
+\usepackage{varwidth}
+\usepackage{xspace}
+\newcommand{\TikZ}{Ti\textit{k}Z\xspace}
+\definecolor{example-color}{gray}{1}
+\definecolor{example-border-color}{gray}{.8}
+\tcbset{
+ coltitle=black,
+ colback=example-color,
+ colframe=example-border-color,
+ enhanced,breakable,
+ pad at break*=1mm,
+ toprule=1.2mm,
+ bottomrule=1.2mm,
+ leftrule=1mm,
+ rightrule=1mm,
+ toprule at break=-1mm,
+ bottomrule at break=-1mm,
+ before upper={\widowpenalties=3 10000 10000 150}
+}
+\tikzset{
+ /Dynkin diagram,
+ edge length=1cm,
+ ordering=Carter,
+ vertical shift=0}
+\tikzset{
+ background rectangle/.style={
+ shade,
+ top color=olive!20,
+ bottom color=white,
+ draw=olive!15,
+ very thick,
+ rounded corners},
+}
+\begin{document}
+\maketitle
+\begin{center}%
+ \begin{tikzpicture}[show background rectangle]
+ \hasse[
+ edge/.style={},
+ root radius=.02cm,
+ edge length=.5cm,
+ edge quotes/.style={opacity=0}%
+ ]{E}{8}%
+ \end{tikzpicture}%
+\end{center}%
+\begin{center}
+\begin{varwidth}{\textwidth}
+\tableofcontents
+\end{varwidth}
+\end{center}
+\setlength{\arrayrulewidth}{1.5pt}
+
+\section{Quick introduction}
+This package draws the Hasse diagram of the poset of the positive simple roots of each complex simple Lie group, as drawn by Ringel \cite{Ringel:2013}.
+\begin{tcolorbox}[title={Load the package}]
+\begin{Verbatim}
+\documentclass{article}
+\usepackage{lie-hasse}
+\begin{document}
+The Hasse diagram of \(F_4\) is
+\begin{center}
+\hasse[edge length=1cm]{F}{4}
+\end{center}
+\end{document}
+\end{Verbatim}
+\end{tcolorbox}
+\par\noindent{}The Hasse diagram of \(F_4\) is
+\begin{center}
+\hasse[edge length=1cm]{F}{4}
+\end{center}
+Each edge is labelled with the simple root by which vertices differ.
+\begin{tcblisting}{title={Inside a \TikZ statement}}
+\(B_4\) has Dynkin diagram \tikz \dynkin[edge length=.35cm]{B}{4};, Hasse diagram
+\begin{center}
+\hasse[edge length=1cm]{B}{4}
+\end{center}
+\end{tcblisting}
+\begin{tcblisting}{title={Inside a Dynkin diagram environment, diagrams fit together}}
+The Hasse diagram of \(B_4\) is
+\begin{dynkinDiagram}[vertical shift=0,edge length=1cm]{B}{4}
+\hasse{B}{4}
+\end{dynkinDiagram}
+\end{tcblisting}
+We shut off the default vertical shift of the Dynkin diagram, so that it starts at the origin.
+There is an option to \verb!\hasse! for this:
+\begin{tcblisting}{title={Attaching the Dynkin diagram}}
+The Hasse diagram of \(B_4\) is
+\begin{center}
+\hasse[attach Dynkin diagram=true]{B}{4}
+\end{center}
+\end{tcblisting}
+Unfortunately, attaching a Dynkin diagram looks terrible for \(D\) or \(E\) series, so a Dynkin diagram appears below.
+\begin{tcblisting}{title={Attaching the Dynkin diagram}}
+The Hasse diagram of \(D_5\) is
+\begin{center}
+\hasse[attach Dynkin diagram=true]{D}{5}
+\end{center}
+\end{tcblisting}
+\begin{tcblisting}{title={Inside a \TikZ environment}}
+\begin{tikzpicture}
+\hasse{A}{4}
+\draw (4;1) circle (5pt);
+\draw[red] (2;3) circle (5pt);
+\end{tikzpicture}
+\end{tcblisting}
+In this example, we see that the roots of the Hasse diagram are \TikZ{} nodes labelled \(g;i\) for grade \(g\) (i.e. \(g\) units up the page) and index \(i\) (i.e. \(i^{\text{th}}\) root of grade \(g\) drawn on the page, starting from the left).
+
+\section{Inherited options}
+The Lie Hasse package inherits options from the Dynkin diagrams package: the edge lengths are set with
+\begin{Verbatim}
+\tikzset{/Dynkin diagram/edge lengths=1.2cm}
+\end{Verbatim}
+and similarly the ordering of roots with
+\begin{Verbatim}
+\tikzset{/Dynkin diagram/ordering=Bourbaki}
+\end{Verbatim}
+
+\section{Prettier}
+The package includes a more elaborate \verb!\hasseDiagrams! command, taking a list of semicolon separated Dynkin diagram identfiers.
+\begin{tcolorbox}[title={With some global options to make prettier diagrams}]
+\begin{Verbatim}
+\tikzset{
+ background rectangle/.style={
+ shade,
+ top color=olive!20,
+ bottom color=white,
+ draw=olive!15,
+ very thick,
+ rounded corners},
+ /Lie Hasse diagram,
+ edge length=1.2cm,
+ show name=true,
+ vertical shift=0}
+\hasseDiagrams{A4;B4;C4}
+\end{Verbatim}
+\end{tcolorbox}
+\begingroup
+\tikzset{
+ background rectangle/.style={
+ shade,
+ top color=olive!20,
+ bottom color=white,
+ draw=olive!15,
+ very thick,
+ rounded corners},
+ /Lie Hasse diagram,
+ edge length=1.2cm,
+ show name=true,
+ vertical shift=0}
+\hasseDiagrams{A4;B4;C4}
+\endgroup
+Global options:
+\begin{verbatim}
+ edge/.style={ultra thick},
+ edge quotes/.style={/Dynkin diagram/text style,auto,inner sep=2pt},
+\end{verbatim}
+allow to change the edges, and to change the way that labels are printed, and how close labels are to the edges.
+
+
+
+\section{Root order}
+We order the roots as in the Dynkin diagram package: with orderings Adams, Bourbaki, Carter, Dynkin and Kac.
+\emph{Warning:} the default is Carter, \emph{not} Bourbaki; the default in the Dynkin diagram package is Bourbaki.
+We can use this like:
+\begin{Verbatim}
+\tikzset{/Lie Hasse diagram,show name=true,show ordering=true}
+\hasseDiagrams{[ordering=Adams]E6;[ordering=Bourbaki]E6}
+\hasseDiagrams{[ordering=Carter]E6;[ordering=Dynkin]E6}
+\hasseDiagrams{[ordering=Kac]E6}
+\end{Verbatim}
+
+\begingroup
+\tikzset{/Lie Hasse diagram,show name=true,show ordering=true}
+\hasseDiagrams{[ordering=Adams]E6;[ordering=Bourbaki]E6}
+\hasseDiagrams{[ordering=Carter]E6;[ordering=Dynkin]E6}
+\hasseDiagrams{[ordering=Kac]E6}
+\endgroup
+
+\section{Graph height and width}
+The \emph{height} of a Hasse diagram is the number of grades.
+The \emph{width} of each grade is the number of vertices on that grade.
+We recover these with
+\begin{Verbatim}
+\newcount\h
+\rootSystemHeight[G][2]{\h}
+\end{Verbatim}
+to store the height of \(G_2\) in a counter called \verb!\h!, and
+\begin{Verbatim}
+\newcount\w
+\rootSystemWidthAtGrade[G][2]{3}{\w}%
+\end{Verbatim}
+to store the width of \(G_2\) at grade \(3\) in a counter called \verb!\w!.
+
+Once you use \verb!\dynkin{G}{2}! or \verb!\hasse{G}{2}! or the other commands, like
+\begin{Verbatim}
+\rootSystemHeight[G][2]{\h}
+\end{Verbatim}
+the system stores that your default root system is \(G_2\).
+Subsequently calls to
+\begin{Verbatim}
+\rootSystemHeight{\h}
+\end{Verbatim}
+and
+\begin{Verbatim}
+\rootSystemWidthAtGrade{3}{\w}
+\end{Verbatim}
+ do not need to specify the root system.
+
+\begingroup
+The \verb!show height! option:
+\begin{Verbatim}
+\tikzset{/Lie Hasse diagram,show name=true,show height=true}
+\hasseDiagrams{G2}
+\end{Verbatim}
+\tikzset{/Lie Hasse diagram,show name=true,show height=true}
+\hasseDiagrams{G2}
+The \verb!show widths! option:
+\begin{Verbatim}
+\tikzset{/Lie Hasse diagram/show widths=true}
+\hasseDiagrams{G2}
+\end{Verbatim}
+\tikzset{/Lie Hasse diagram/show widths=true}
+\hasseDiagrams{G2}
+\tikzset{/Lie Hasse diagram/show height=false}
+\tikzset{/Lie Hasse diagram/show widths=false}
+\endgroup
+
+\section{Root decompositions}
+Each positive root in a root system is a unique nonnegative integer linear combination of positive simple roots.
+We can recover this expression as
+\begin{Verbatim}
+\rootSum[G][2]{5}{1}{\rs}
+\end{Verbatim}
+which, for the root system \(G_2\), and the root at position \(5;1\) in our Hasse diagram, stores in the variable \verb!\rs! a string which looks like \rootSum[G][2]{5}{1}{\rs}\texttt{\rs}.
+This is a comma separated list of the integer coefficients.
+\emph{Warning:} for the moment, this list of coefficients is in Carter ordering.
+If we omit \verb![G][2]!, the current default root system is implied.
+
+Here is the Dynkin diagram of \(E_8\), indicating the order of the roots in Carter ordering.
+\begin{Verbatim}
+\dynkin[label,ordering=Carter,edge length=.35cm]{E}{8}
+\end{Verbatim}
+\begin{center}
+\dynkin[label,ordering=Carter,edge length=.35cm]{E}{8}
+\end{center}
+Here is the same Dynkin diagram, except showing, at each simple root, the coefficient of that simple root in the highest root.
+\begin{Verbatim}
+\rootSum[E][8]{29}{1}{\rs}
+\dynkin[labels=\rs,ordering=Carter,edge length=.35cm]{E}{8}
+\end{Verbatim}
+\rootSum[E][8]{29}{1}{\rs}
+\begin{center}
+\dynkin[labels=\rs,ordering=Carter,edge length=.35cm]{E}{8}
+\end{center}
+
+The option \verb!for all roots! allows execution of code once on every root.
+\begin{Verbatim}
+\tikzset{/Lie Hasse diagram,
+ edge length=3.2cm,
+ compact root/.code={},
+ noncompact root/.code={},
+ edge quotes/.style={opacity=0},
+ embedded Dynkin diagram/.style={
+ edge length=.4cm,
+ root radius=.05cm
+ },
+ for all roots/.code 2 args={\drawRootAsDynkinSum{#1}{#2}}}
+\hasseDiagrams{D5}
+\end{Verbatim}
+\begingroup
+\tikzset{/Lie Hasse diagram,
+ edge length=3.2cm,
+ compact root/.code={},
+ noncompact root/.code={},
+ edge quotes/.style={opacity=0},
+ embedded Dynkin diagram/.style={
+ edge length=.4cm,
+ root radius=.05cm
+ },
+ for all roots/.code 2 args={\drawRootAsDynkinSum{#1}{#2}}}
+\hasseDiagrams{D5}
+\endgroup
+See more below on compact versus noncompact roots; the code \verb!compact! is applied to draw all of the compact roots, and the code \verb!noncompact! to draw the noncompact roots.
+Setting those codes to be empty, and setting \verb!edge quotes! to be transparent, we get a much simpler Hasse diagram, so that we can see the embedded Dynkin diagrams more clearly.
+
+\section{\texorpdfstring{For all roots \ldots}{For all roots ...}}
+You can make your own macros loop over all of the roots: you define a macro \verb!\foo{g}{i}!, which is fed the grade \(g\) of each root in the diagram, and the \emph{index} \(i\).
+A simple example:
+\begin{Verbatim}
+\newcommand{\foo}[2]%
+{%
+ \node[below,scale=.5] at (#1;#2) {\(#1,#2\)};%
+}%
+\end{Verbatim}
+\newcommand{\foo}[2]%
+{%
+ \node[below,scale=.75] at (#1;#2) {\(#1,#2\)};%
+}%
+Inside a \TikZ{} or \verb!dynkinDiagram! environment:
+\begin{Verbatim}
+\tikzset{/Lie Hasse diagram/edge quotes/.style={opacity=0},
+ /Dynkin diagram/edge length=1.5cm}
+\begin{tikzpicture}
+ \hasse{D}{6}%
+ \forAllPositiveRootsInHasseDiagram{\foo}%
+\end{tikzpicture}
+\end{Verbatim}
+\begingroup
+\tikzset{/Lie Hasse diagram/edge quotes/.style={opacity=0},
+ /Dynkin diagram/edge length=1.5cm}
+\begin{tikzpicture}
+ \hasse{D}{6}%
+ \forAllPositiveRootsInHasseDiagram{\foo}%
+\end{tikzpicture}
+
+If you put this into the \verb!for all roots! option, it executes on its own:
+\begin{Verbatim}
+\tikzset{/Lie Hasse diagram/for all roots/.code 2 args={\foo{#1}{#2}}}
+\hasseDiagrams{C4;D4}
+\end{Verbatim}
+\begingroup
+\tikzset{/Lie Hasse diagram/for all roots/.code 2 args={\foo{#1}{#2}}}
+\hasseDiagrams{C4;D4}
+\endgroup
+\endgroup
+
+\section{Three dimensional effect}
+We draw the \(D,E,F\) Hasse diagrams, following Ringel \cite{Ringel:2013}, as an arrangement of cubes.
+Nutma \cite{Nutma:2010} draws the Hasse diagrams using a more elementary approach, but including also the affine Kac--Moody algebras.
+Opposite sides of any square have the same edge label, by commutativity of addition.
+Hence we don't need to see every edge perfectly.
+The three dimensional effect is the default:
+\begin{Verbatim}
+\hasse{D}{4}\hasse{E}{6}
+\end{Verbatim}
+\begin{center}
+\hasse{D}{4}\hasse{E}{6}
+\end{center}
+We can turn it off:
+\begin{Verbatim}
+\hasse[three D=false]{D}{4}
+\hasse[three D=false]{E}{6}
+\end{Verbatim}
+\begin{center}
+\hasse[three D=false]{D}{4}
+\hasse[three D=false]{E}{6}
+\end{center}
+or globally with \verb!\tikzset{/Lie Hasse diagram/three D=false}!.
+
+The astute reader will perhaps notice that the three dimensional effect is not realistic.
+To be Hasse diagrams, the roots have to line up horizontally by grade.
+This is inconsistent with three dimensional projection of our cubes.
+We have also tried to use only a small number of layers in the three dimensional geometry, so the images are not perfect, but easy enough to read.
+
+We can change the \verb!z shift! to slant the three dimensional images to the right:
+\begingroup
+\begin{Verbatim}
+\hasse[z shift=.1]F4\hasse[z shift=.2]F4\hasse[z shift=.3]F4\hasse[z shift=.4]F4
+\end{Verbatim}
+\hasse[z shift=.1]F4\hasse[z shift=.2]F4\hasse[z shift=.3]F4\hasse[z shift=.4]F4
+\endgroup
+
+We only use three colours and opacities for the faces:
+\begin{Verbatim}
+ top/.style={black!20,opacity=.4},
+ left/.style={black!20,opacity=.9},
+ right/.style={black!20,opacity=.6},
+\end{Verbatim}
+You can change these:
+\begin{Verbatim}
+\hasse[
+ top/.style={red,opacity=.1},
+ right/.style={red,opacity=.2},
+ left/.style={red,opacity=.4}]E6
+\end{Verbatim}
+\begin{center}
+\hasse[
+ top/.style={red,opacity=.1},
+ right/.style={red,opacity=.2},
+ left/.style={red,opacity=.4}]E6
+\end{center}
+
+\section{Label the simple roots}
+Ringel \cite{Ringel:2013} labels his edges like
+\begin{Verbatim}
+\hasseDiagrams{[labels={f,e,d,c,u,b,a}]E7}
+\end{Verbatim}
+\hasseDiagrams{[labels={f,e,d,c,u,b,a}]E7}
+
+\section{Parabolic subgroups}
+This package offers nothing over Ringel's original pictures, except that the user can pick some simple roots whose associated edges are drawn differently.
+The chosen simple roots are called \emph{compact}, following terminology from the theory of parabolic subgroups.
+We let the reader explore the notation for parabolic subgroups in the Dynkin diagrams package, and use this to declare various roots compact.
+\begin{Verbatim}
+\tikzset{/Lie Hasse diagram,attach Dynkin diagram=true,three D=false}
+\hasseDiagrams{D{**x*x*x*}}
+\end{Verbatim}
+\begingroup
+\tikzset{/Lie Hasse diagram,attach Dynkin diagram=true,three D=false}
+\hasseDiagrams{D{**x*x*x*}}
+\endgroup
+Our motivation comes from trying to identify the invariant vector subbundles of the tangent bundle of a rational homogeneous variety \cite{MathOverflow:123801}.
+Such diagrams are often unreadable if we don't turn off the three dimensional graphics.
+By default, noncompact root edges are not drawn.
+\begingroup
+\tikzset{/Lie Hasse diagram,attach Dynkin diagram=true,show name=false,three D=false}
+\begin{Verbatim}
+\hasseDiagrams{E{*xx*x*}}
+\end{Verbatim}
+\hasseDiagrams{E{*xx*x*}}
+\begin{Verbatim}
+\hasseDiagrams{A{x*x*}}
+\end{Verbatim}
+\hasseDiagrams{A{x*x*}}
+\begin{Verbatim}
+\hasseDiagrams{[parabolic=113]B8}
+\end{Verbatim}
+\hasseDiagrams{[parabolic=113]B8}
+\begin{Verbatim}
+\hasseDiagrams{C{**xx*x**}}
+\end{Verbatim}
+\hasseDiagrams{C{**xx*x**}}
+\newpage
+\begin{Verbatim}
+\hasseDiagrams{E{*x*x*x**}}
+\end{Verbatim}
+\hasseDiagrams{E{*x*x*x**}}
+\newpage
+\begin{Verbatim}
+\hasseDiagrams{F{**xx}}
+\end{Verbatim}
+\hasseDiagrams{F{**xx}}
+\begin{Verbatim}
+\hasseDiagrams{G{*x}}
+\end{Verbatim}
+\hasseDiagrams{G{*x}}
+\endgroup
+
+\section{Examples}
+\begingroup
+\tikzset{/Lie Hasse diagram,attach Dynkin diagram=true,show name=true}
+\begin{Verbatim}
+\hasseDiagrams{A1;A2;A3;A4;A5;A6}
+\hasseDiagrams{B3;B4;B5}
+\hasseDiagrams{C2;C3;C4}
+\hasseDiagrams{C5;C6}
+\hasseDiagrams{E6;E7}
+\hasseDiagrams{E8}
+\hasseDiagrams{F4;G2}
+\end{Verbatim}
+\hasseDiagrams{A1;A2;A3;A4;A5;A6}
+\hasseDiagrams{B3;B4;B5}
+\hasseDiagrams{C2;C3;C4}
+\hasseDiagrams{C5;C6}
+\hasseDiagrams{E6;E7}
+\hasseDiagrams{E8}
+\hasseDiagrams{F4;G2}
+\endgroup
+
+\section{Black and white}
+Publishing in colour on paper can be expensive.
+Simple global options:
+\begin{Verbatim}
+\tikzset{
+ background rectangle/.style={
+ shade,
+ top color=gray!15,
+ bottom color=white,
+ draw=gray!5,
+ very thick,
+ rounded corners},
+ /Dynkin diagram/text style/.style={black,scale=.75},
+ /Lie Hasse diagram,
+ edge length=1cm,
+ edge/.style={draw=black!50,ultra thick},
+ edge quotes/.style={black,auto,inner sep=3pt,scale=.75},
+ three D=true,
+ show name=true}
+\end{Verbatim}
+\begingroup
+\tikzset{
+ background rectangle/.style={
+ shade,
+ top color=gray!15,
+ bottom color=white,
+ draw=gray!5,
+ very thick,
+ rounded corners},
+ /Dynkin diagram/text style/.style={black,scale=.75},
+ /Lie Hasse diagram,
+ edge length=1cm,
+ edge/.style={draw=black!50,ultra thick},
+ edge quotes/.style={black,auto,inner sep=3pt,scale=.75},
+ three D=true,
+ show name=true}%
+change our examples to
+\hasseDiagrams{A1;A2;A3;A4;A5;A6}
+\hasseDiagrams{B3;B4;B5}
+\hasseDiagrams{C2;C3;C4}
+\hasseDiagrams{C5;C6}
+\hasseDiagrams{E6;E7}
+\hasseDiagrams{E8}
+\hasseDiagrams{F4;G2}
+\endgroup
+
+\bibliographystyle{amsplain}
+\bibliography{lie-hasse}
+\end{document}